Hello I got this problem, if $p$ is prime then the polynomial: $$q(x)=1+x+x^2+...+x^{p-1} $$ is irreductible in $\mathbb{Q}[x]$. I don't know how to do this exactly. And a problem is that a condition to this question is that I can't use the Eisenstein Criterion, only the basic things about ring theory. Thanks if you can help me with this.
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IIRC, Serge Lang's Algebra textbook has a different proof not using Eisenstein's criterion (more useful for proving the general case of cyclotomic polynomials are irreducible, but should also work OK for this case). – Daniel Schepler Nov 02 '17 at 17:21
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But in this case I cant use Eisenstein... I am looking for other way to do it. – TeemoJg Nov 02 '17 at 17:22
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You can do the same method, and just not refer to the name "Eisenstein". It is completely elementary, and the best way, I think. – Dietrich Burde Nov 02 '17 at 17:23
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I have no idea, why this marked as a duplicate, when the OP explicitly states, that he does not want to use Eisenstein. You deny someone the chance to give a proof without Eisenstein. There are plenty of them. – MooS Nov 03 '17 at 09:54